Order notes (private note) are only available for backend when using get_comments() function.
If you look at WC_Comments exclude_order_comments() method you will see that front end queries are filtered regarding private order notes…

So the turn around is to build a custom function to get the private Order notes:

function get_private_order_notes( $order_id){
    global $wpdb;

    $table_perfixed = $wpdb->prefix . 'comments';
    $results = $wpdb->get_results("
        SELECT *
        FROM $table_perfixed
        WHERE  `comment_post_ID` = $order_id
        AND  `comment_type` LIKE  'order_note'
    ");

    foreach($results as $note){
        $order_note[]  = array(
            'note_id'      => $note->comment_ID,
            'note_date'    => $note->comment_date,
            'note_author'  => $note->comment_author,
            'note_content' => $note->comment_content,
        );
    }
    return $order_note;
}

Code goes in function.php file of your active child theme (or theme) or also in any plugin file.

This code is tested and works.

Usage (for example the $order_id = 6238 ):

$order_id = 6238;
$order_notes = get_private_order_notes( $order_id );
foreach($order_notes as $note){
    $note_id = $note['note_id'];
    $note_date = $note['note_date'];
    $note_author = $note['note_author'];
    $note_content = $note['note_content'];

    // Outputting each note content for the order
    echo '<p>'.$note_content.'</p>';
}

There woo documentation https://docs.woocommerce.com/wc-apidocs/function-wc_get_order_notes.html

Example:

wc_get_order_notes([
   'order_id' => $order->get_id(),
   'type' => 'customer',
]);

Result

Array
(
    [0] => stdClass Object
        (
            [id] => 11
            [date_created] => WC_DateTime Object
                (
                    [utc_offset:protected] => 3600
                    [date] => 2019-03-21 11:38:51.000000
                    [timezone_type] => 1
                    [timezone] => +00:00
                )

            [content] => Hi, blah blah blah
            [customer_note] => 1
            [added_by] => admin
        )

)

There is alternative best way for getting order notes.

/**
 * Get all approved WooCommerce order notes.
 *
 * @param  int|string $order_id The order ID.
 * @return array      $notes    The order notes, or an empty array if none.
 */
function custom_get_order_notes( $order_id ) {
    remove_filter( 'comments_clauses', array( 'WC_Comments', 'exclude_order_comments' ) );
    $comments = get_comments( array(
        'post_id' => $order_id,
        'orderby' => 'comment_ID',
        'order'   => 'DESC',
        'approve' => 'approve',
        'type'    => 'order_note',
    ) );
    $notes = wp_list_pluck( $comments, 'comment_content' );
    add_filter( 'comments_clauses', array( 'WC_Comments', 'exclude_order_comments' ) );
    return $notes;
}